The two most relevant attempts of his that added torsion are the Einstein-Cartan theory which hybridizes curvature and torsion, and Teleparallel Gravity which replaces curvature with torsion entirely. 

 https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory

https://en.wikipedia.org/wiki/Teleparallelism

I’m betting on a beautiful variation of EC.